Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 672531, 15 pages

doi:10.1155/2012/672531

Research Article

Hyers-Ulam-Rassias RNS Approximation of Euler-Lagrange-Type Additive Mappings

H. Azadi Kenary,¹ H. Rezaei,¹ A. Ebadian,² and A. R. Zohdi³

¹Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran

²Department of Mathematics, Payame Noor University, Tehran, Iran

³Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

Received 24 December 2011; Revised 5 March 2012; Accepted 19 March 2012

Academic Editor: Tadeusz Kaczorek

Copyright © 2012 H. Azadi Kenary et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Recently the generalized Hyers-Ulam (or Hyers-Ulam-Rassias) stability of the following functional equation where , proved in Banach modules over a unital -algebra. It was shown that if , for some and a mapping satisfies the above mentioned functional equation then the mapping is Cauchy additive. In this paper we prove the Hyers-Ulam-Rassias stability of the above mentioned functional equation in random normed spaces (briefly RNS).

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

The paper of Rassias has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability of functional equations. In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [5] by replacing the bound by a general control function .

The functional equation:

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings , where is a normed space and is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability of the quadratic functional equation.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2, 4, 5, 9–28]).

In the sequel, we will adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in [29]. Throughout this paper, the spaces of all probability distribution functions are denoted by . Elements of are functions , such that is left continuous and nondecreasing on , and . It’s clear that the subset , where , is a subset of . The space is partially ordered by the usual point-wise ordering of functions, that is, for all , if and only if . For every , is the element of defined by

One can easily show that the maximal element for in this order is the distribution function .

Definition 1.1. A function is a continuous triangular norm (briefly a -norm) if satisfies the following conditions:

(i) is commutative and associative;(ii) is continuous;(iii) for all ;(iv) whenever and for all .

Three typical examples of continuous -norms are , and . Recall that, if is a -norm and is a given of numbers in , is defined recursively by and for .

Definition 1.2. A random normed space (briefly ) is a triple , where is a vector space, is a continuous -norm, and is a mapping such that the following conditions hold.

(i) for all if and only if .(ii) for all , , and .(iii), for all and .

Definition 1.3. Let be an RNS.

(i)A sequence in is said to be convergent to in if for all , (ii)A sequence in is said to be Cauchy sequence in if for all ,.(iii)The -space is said to be complete if every Cauchy sequence in is convergent.

Theorem 1.4. If is RNS and is a sequence such that , then .

In this paper, we investigate the generalized Hyers-Ulam stability of the following additive functional equation of Euler-Lagrange type:

where , , and for some , in random normed spaces.

Every solution of the functional equation (1.3) is said to be a generalized Euler-Lagrange type additive mapping.

2. RNS Approximation of Functional Equation (1.3)

Remark 2.1. Throughout this paper, will be real numbers such that for fixed .

Theorem 2.2. Let be a real linear space, be an RN space, be a function such that for some ,

and for all and

Let be a complete RN space. If is a mapping such that for all and

then there is a unique generalized Euler-Lagrange-type additive mapping such that, for all and all

Proof. For each with , let in (2.3). Then we get the following inequality:

for all , where

for all and all , and

Letting in (2.5), we get

for all . Similarly, letting in (2.5), we get

for all . It follows from (2.5), (2.8), and (2.9) that for all

Replacing and by and in (2.10), we get that

for all . Putting in (2.11), we get

for all . Replacing and by and in (2.11), respectively, we get

for all . It follows from (2.12) and (2.13) that

for all . So

Replacing by in (2.15) and using (2.1), we get

for all and all . Therefore, we have

for all . This implies that

Replacing by in (2.18), we obtain

Since the right-hand side of the above inequality tends to 1, when , then the sequence is a Cauchy sequence in complete RN space (), so there exists some point such that

for all .

Fix and put in (2.19). Then we obtain

and so, for every , we have

Taking the limit as and using (2.22), we get

Since was arbitrary by taking in (2.23), we get

Replacing by for all , in (2.3), we get for all and for all ,

since

We conclude that

To prove the uniqueness of mapping , assume that there exists another mapping which satisfies (2.4). Fix , clearly and , for all . Since , so

Since the right-hand side of the above inequality tends to 1, when , therefore, it follows that for all , and so . This completes the proof.

Corollary 2.3. Let be a real linear space, be an RN space, and a complete RN space. Let , and be a mapping with and satisfying

for all and . Then the limit exists for all and defines a unique Euler-Lagrange additive mapping such that

for all and .

Proof. Let and be defined as .

Corollary 2.4. Let be a real linear space, be an RN space, and a complete RN space. Let and be a mapping with and satisfying

for all for all and all . Then, the limit exists for all and defines a unique Euler-Lagrange additive mapping such that

for all and .

Proof. Let and be defined as .

Theorem 2.5. Let be a real linear space, be an RN space, be a function such that for some ,

and for all and , . Let be a complete RN space. If is a mapping satisfying (2.3), then there is a unique generalized Euler-Lagrange-type additive mapping such that, for all

for all and all .

Proof. Replacing by in (2.14) and using (2.33), we obtain

So

for all . This implies that

Proceeding as in the proof of Theorem 2.2, one can easily show that the sequence is a Cauchy sequence in complete RN space , so there exists some point such that

for all .

Taking the limit from both sides of the above inequality, we obtain (2.34).

The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.6. Let be a real linear space, be an RN space and a complete RN space. Let , and be a mapping with and satisfying

for all for all and all . Then the limit exists for all and defines a unique Euler-Lagrange additive mapping such that

for all and .

Proof. Let and be defined as .

Corollary 2.7. Let be a real linear space, be an RN space and a complete RN space. Let and be a mapping with and satisfying

for all and . Then, the limit exists for all and defines a unique Euler-Lagrange additive mapping such that

for all and .

Proof. Let and be defined as .

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